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13.42 Homework #3 Spring 2005 Out: Thursday, February 17, 2005 Due: Thursday, February 24, 2005 Problem 1: Probability Review Problem. Two fair dice are rolled simultaneously with the following defined events: A = {Sum of roll is even} B = {Sum of roll is odd} C = {Doubles are rolled} D = {Sum of roll is ≤ 8 } Find the following probabilities: a) P( A ∪ D) b) P( B ∩ C ) c) P(C ∪ D) d) P ( B ∩ D) e) P ( A ∩ C ) Problem 2: Probability Review Problem. Find the probability of drawing a full house in a five-card hand. (Assume that every possible five-card hand drawn from a standard deck of 52 cards has the same probability of being selected.) Problem 3: Let random variable X be the total sum of the outcomes of 3 flips of a fair coin when the value of 1 is assigned to heads and the value of 0 is assigned to tails. a) Find the probability mass function (i.e. the discrete version of the probability density function), p X ( x) = P( X = x) . b) Find the cumulative distribution function, FX ( x) = P( X ≤ x) . c) Using the probability function, find the mean, variance, and standard deviation. Problem 4: Let the random variable X have a cumulative distribution function FX(x): ⎧0 ⎪1 ⎪ x+ 1 4 ⎪4 ⎪⎪ 1 FX ( x) = ⎨ ⎪4 ⎪ 2 ⎪x ⎪ ⎪⎩1 if x < −1 if if if if ⎫ ⎪ − 1 ≤ x < 0⎪ ⎪ 1 ⎪⎪ 0≤ x< ⎬ 2 ⎪ 1 ⎪ ≤ x <1 ⎪ 2 ⎪ 1≤ x ⎪⎭ Find the following probabilities: 1 a) P( x ≤ ) 2 b) P( x ≥ 0) 1 c) P(0 ≤ x < ) 2 3 d) P( x ≤ ) 4 e) P( x ≥ 1) Problem 5: Let Y be a random variable with the following probability density function: ⎧2 x f X ( x) = ⎨ ⎩0 if 0 ≤ x < 1⎫ ⎬ otherwise ⎭ a) Find the expected value, μ X . b) Find the standard deviation, σ X . Problem 6: Consider a random process, wave elevation: η (t ) = A cos(ωt + φ ) where ω and φ are constants and A is a Gaussian random variable with zero mean. a) What is the probability density function of A? b) What is the dynamic pressure, pd(t), under this wave elevation? c) Determine whether pd(t) is a stationary process. Problem 7: ( MATLAB recommended for this problem.) The random wave elevation at a given point in the ocean may be represented as follows: N η (t ) = ∑ Ai cos(ωi t + φi ) i =1 Where Ai , ωi , and φi are the wave amplitude, frequency, and random phase angle (with uniform distribution from 0 to 2π) of wave component number i. The amplitudes Ai corresponding to each frequency ωi may be found from a known wave energy spectrum, Sη (ω ) , where Sη (ω )δω = 1 2 Ai . 2 a) Using the wave energy spectrum given in Figure 1, generate 10 realizations of the random process η (t ) . Let t = [0, 90] seconds. (Hint: Use the MATLAB function rand to generate a realization of the random variable φi .) b) Compute the ensemble averages (mean and variance at t = 30 seconds, and compute the temporal averages (mean and variance) for each realization. Compare the results. Problem 8: Give short answers to the following questions. a) What causes the vast majority of sea waves? b) Why do storm waves have a limiting wavelength? c) What is significant wave height? d) Why is significant wave height used so extensively in design practices?